Theoretical and experimental determination of mechanical ...

OAK RIDGE NATIONAL LABORATORY LIBRAE ORNL/TM-5331

3 MM5b DDS370fi 1Cy ff

Theoretical and ExperimentalDetermination of Mechanical Properties

of Superconducting Composite Wire

UCN-7969

(3 3-67)

W. H. GrayC. T. Sun

UTO

ARY

COPYl PERSON

i this

K RIDGE NATIONAL. LABORATO

CENTRAL RESEARCH LIBRARY

DOCUMENT COLLECTION

IBRARY LOAN COPDO NOT TRANSFER TO ANOTHER PERSON

If you wish someone else to see this

document, send in name with documentand the library will arrange a loan.

OAK RIDGE NATIONAL LABORATORYOPERATED BY UNION CARBIDE CORPORATION FOR THE ENERGY RESEARCH AND DEVELOPMENT ADMINISTRATION

Printed in the United States of America. Available from

National Technical Information Service

U.S. Department of Commerce

5285 Port Royal Road, Springfield, Virginia 22161

Price: Printed Copy $4.50; Microfiche $2.25

This report was prepared as an account of work sponsored by the United States

Government. Neither the United States nor the Energy Research and DevelopmentAdministration/United States Nuclear Regulatory Commission, nor any of their

employees, nor any of their contractors, subcontractors, or their employees, makes

any warranty, express or implied, or assumes any legal liability or responsibility forthe accuracy, completeness or usefulness of any information, apparatus, product or

process disclosed, or represents that its use would not infringe privately owned rights.

ORNL/TM-5331

Contract No. W-7405-eng-26

THEORETICAL AND EXPERIMENTAL DETERMINATION OF MECHANICAL

PROPERTIES OF SUPERCONDUCTING COMPOSITE WIRE

W. H. Gray

Superconducting Magnet Development ProgramThermonuclear Division

and

C. T. Sun

Iowa State University

JULY 1976

NOTICE This document contains information of a preliminary nature

and was prepared primarily for internal use at the Oak Ridge National

Laboratory. It is subject to revision or correction and therefore does

not represent a final report.

OAK RIDGE NATIONAL LABORATORY

Oak Ridge, Tennessee 37830operated by

UNION CARBIDE CORPORATION

for the

ENERGY RESEARCH AND DEVELOPMENT ADMINISTRATION

OAK RIDGE NATIONAL LABORATORY LIBRARY

3 MM5b DDE37Dfl 1

TABLE OF CONTENTS

ABSTRACT 1

INTRODUCTION 1

THEORETICAL INVESTIGATION 2

1.1 Self-Consistent Model Methods 2

1.2 Variational Methods 3

1.3 Exact Methods 3

1.4 Mechanics of Materials Method 3

1.5 The Halpin-Tsai Equations 4

EXPERIMENTAL DETERMINATION OF ELASTIC CONSTANTS 5

2.1 Experimental Determination of E., and V^2 6

2.2 Experimental Determination of E22 and G2„ 6

2.3 Experimental Determination of the LongitudinalShear Modulus G.. „ 7

CONCLUSIONS 8

REFERENCES 10

NOMENCLATURE 12

APPENDIX I 13

APPENDIX II 23

APPENDIX III 25

ACKNOWLEDGMENTS 35

FIGURE CAPTIONS 36

in

THEORETICAL AND EXPERIMENTAL DETERMINATION OF MECHANICAL

PROPERTIES OF SUPERCONDUCTING COMPOSITE WIRE*

W. H. Gray and C. T. Sunt

ABSTRACT

The object of this research is to characterize the mechanical

properties of a composite superconducting (NbTi/Cu) wire in terms

of the mechanical properties of each constituent material. For a

particular composite superconducting wire, five elastic material

constants were experimentally determined and theoretically calculated.

Since the Poisson's ratios for the fiber and the matrix material

were very close, there was essentially no (less than 1%) difference

among all the theoretical predictions for any individual mechanical

constant. Because of the expense and difficulty of producing elastic

constant data of 0.1% accuracy, and therefore conclusively determining

which theory is best, no further experiments were performed.

INTRODUCTION

The object of this research is to characterize the mechanical

properties of a composite superconducting (NbTi/Cu) wire in terms of

the mechanical properties of each constituent material. In 1973, the

Cryogenics Division of the National Bureau of Standards (NBS) published

an interim report in which a preliminary investigation of the mechan

ical properties of a solenoid coil composite was made. The coil inves

tigated consisted of epoxy, fiberglass, and composite superconducting

wire. Both theoretical and experimental elastic constants were tabulated

* This research was sponsored in part by the Engineering ResearchInstitute, Iowa State University, Ames, IA 50011, and in partby the Energy Research and Development Administration undercontract with Union Carbide Corporation.

t Department of Engineering Sciences and Mechanics and EngineeringResearch Institute, Iowa State University, Ames, IA 50011.

for a typical piece of coil cut out of a small solenoid. Our report

differs from this NBS work in that we consider only the mechanical

properties of an individual composite superconducting wire.

The theoretical predictions and the experimental procedures to

determine the effective elastic constants of the composite wire are

described in the next two sections of this report.

THEORETICAL INVESTIGATION

Most of the analytical work for predicting the mechanical and

thermal properties of fiber-reinforced composites in terms of volu

metric composition, geometrical arrangement of the fibers, and con

stituent material properties was done before 1970. There are five

approaches to predict the micromechanical behavior of fiber-reinforced

composites.* The essential characteristic of each is described below.

1.1 SELF-CONSISTENT MODEL METHODS

This method was originally proposed by Hershey and Kroner for

crystal aggregates, and was first employed by Hill to derive expres

sions for elastic constants. Hill modeled the composite as a single

fiber embedded in an unbounded macroscopically homogeneous medium,

subjected to a uniform loading at infinity. This uniform loading

produces a uniform strain field in the filament which is then used

to estimate the elastic constants. A similar model proposed by Frohlich

and Sack for predicting the viscosity of a Newtonian fluid containing

a dispersion of equal elastic spheres consists of three concentric

cylinders, the outer one being unbounded. The innermost cylinder is

assumed to have the elastic properties of the filaments; the middle

one has the properties of the matrix; and the outermost has the properties

of the composite. The solid is subjected to homogeneous stresses at

infinity. The resulting elastic fields are determined, and then are

* No attempt is made here to give a comprehensive literature surveyregarding this subject. More references can be found in Ref . 2.

employed to predict the elastic constants of the composite. Applications

of the self-consistent model methods can be found, for example, in

Refs. 7-9.

1.2 VARIATIONAL METHODS

In this method, the energy theorems of classical elasticity are

used to obtain bounds on the mechanical and physical properties of

filamentary composites. The minimum complementary energy theorem

yields a lower bound, while the minimum potential energy theorem yields

the upper bound. Using this approach, bounds for the elastic and

thermal properties of composites have been obtained by many investiga-10-12

tors.

1.3 EXACT METHODS

By assuming that the fibers are arranged in a doubly periodic

rectangular array, a fundamental or repeating element can be estab

lished. The resulting elasticity problem can then be solved either

by introducing a stress function using a series development, or by

numerical techniques such as finite difference or finite element

methods. Once the problem is solved elastically, the resulting elastic

fields can be averaged to get expressions for the desired elastic con

stants. Typical applications of this method can be found in Refs.

13-18.

1.4 MECHANICS OF MATERIALS METHOD

By making simplifying assumptions regarding the mechanical or

thermal behavior of a composite material, the mechanics-of-materials

expressions for the equivalent elastic or thermal constants of unidi-

rectionally reinforced fibrous composite materials can be derived. For

example, to determine the longitudinal Young's modulus, one assumes

that the longitudinal strains in both the matrix and the fiber are the

same; in order to determine the transverse Young's modulus, one assumes

that transverse stresses in both materials are the same. This approach

usually is referred to as the "rule of mixtures." The "rule of mixtures"

expressions for elastic moduli and thermal conductivities can be found in

Refs. 19-21.

1.5 THE HALPIN-TSAI EQUATIONS

For designers, it is often necessary to have simple and rapid com

putational procedures for estimating the macromechanical properties of

a fibrous composite. Such empirical formulas have been developed by22Halpin and Tsai based upon modifications of the results discussed

under approaches 1.1 and 1.3. By estimating the value of a factor

which depends on the geometry of the inclusions, spacing geometry, and

loading conditions, the composite elastic moduli can be approximated.

Reliable estimates for this factor can be obtained by comparing the

Halpin-Tsai equation with the numerical micromechanics solutions. If

used appropriately, the Halpin-Tsai equation can yield very reliable

results without elaborate calculations.

All the above methods make the following three basic assumptions:

(1) each constituent material behaves linearly elastically, (2) the fibers

are straight (without twist), and (3) there are no residual stresses.

For this investigation, we use several of the available theoretical equa

tions to predict the effective elastic mechanical properties of supercon

ducting composite wire. These equations are listed in Appendix I.

In general, a superconducting composite wire may be twisted to

minimize ac power losses in a superconducting magnet. Twisting of the

wire violates an assumption implicit in the derivation of all the

equations presented in Appendix I. However, we believe that this effect

is small (see Appendix II), and for engineering purposes can be neglected.

Other effects, such as inelastic behavior of the wire at higher loading

levels, and residual stresses in the wire introduced during fabrication,

may influence the results presented in this paper, and should be ana

lyzed more thoroughly.

EXPERIMENTAL DETERMINATION OF ELASTIC CONSTANTS

Assuming the NbTi/Cu wire behaves like a transversely isotropic

material, there are five elastic constants of significance. These

constants are: (1) Young's modulus along the direction of the fiber,

ET or E,, (longitudinal Young's modulus), (2) major Poisson's ratio,Li J_X

V or V.,, (3) Young's modulus along the direction normal to the fiber,

ET or E„9 (transverse Young's modulus), (4) minor Poisson's ratio, VTor V9„, and (5) longitudinal shear modulus, G or G^. Since the material properties in a plane normal to the fiber direction are assumed to

be isotropic, the transverse shear modulus G„ or G2„ can be determined

from the isotropic relation

E

G T (1)T 2(1 + VT)

The particular superconducting composite wire chosen for our experi

ment was KRYO-210. This conductor has cross-sectional dimensions of

10.16 mm by 5.08 mm with a copper matrix containing 2640 Nb-45 wt % Ti

superconducting filaments. The copper to superconductor ratio is 6.

The elastic constants which were used for the comparisons are tabulated

below. All elastic constant measurements were made at room temperature.

Young's Modulus Poisson's Ratio Shear ModulusMaterial (GPa) (GPa)

Cu 123 0.345 45.7

NbTi 84 0.33 31.5

* Registered trademark of Magnetics Corporation of America (MCA).

t Tradenames of material are used in this report for clarity. In nocase does such selection imply recommendations or endorsement by theauthors, nor does it imply that the material is necessarily the bestavailable for the purpose.

2.1 EXPERIMENTAL DETERMINATION OF E., AND V..

E,, and V-. can be determined from a simple tension test (see

Fig. 1). The direction of loading is parallel to the fibers of the

conductor. The longitudinal Young's modulus is determined from a a1

vs e, diagram where o\ is equal to the applied load divided by the

cross sectional area of the specimen, and £, is the strain along the

fiber direction of the conductor. Figure 2 represents an experimentally

determined plot of this diagram for KRYO-210 superconductor showing a

value for E^ of 119 GPa. A comparison of the theoretical prediction

and the experimental data is shown in Fig. 3 in this graph, as well as

the four normalized comparison graphs which follow; the legend refers

to the theoretical equations presented in Appendix I. Both equations

predict the same behavior, which deviates approximately 3% from the

experimental data.

The major Poisson's ratio is determined from the slope of the £

vs £ diagram during the same experiment, where £„ is the strain in

either transverse direction. The experimentally determined value was

0.347 (see Fig. 4). The normalized plot (see Fig. 5) showing the

comparison between theoretical prediction and experimental value again

demonstrates little difference between theories with the experimental

data differing from the predictions by about 2%.

2.2 EXPERIMENTAL DETERMINATION OF E 2 AND G„,,

E22 and Gô are determined from a test similar to that described in

2.1 (see Fig. 6). The direction of the applied load is normal to the

fiber axis. The transverse Young's modulus is determined from the a„

vs £2 diagram where o^ is the stress, and £„ is the strain in the direc

tion of the applied force. Figure 7 represents an experimentally deter

mined plot of this diagram for KRYO-210 superconductor showing a value

for E22 of 122 GPa. The experimental data are compared to the theoretical

predictions for E22 in Figure 8. An error of approximately 5% is observed.

Analogously the minor Poisson's ratio, and therefore G „, is deter

mined from an £3 vs £2 diagram. The experimental value for G _ is 43.1

GPa (see Fig. 9). Figure 10 compares this data point with the

theoretical predictions. An error of approximately 2% is observed.

2.3 EXPERIMENTAL DETERMINATION OF THE LONGITUDINAL SHEAR MODULUS G12

The simplest way to determine the longitudinal shear modulus G^2

is to use a tensile specimen with the fibers oriented at a 45 direction24to the geometrical axis of the specimen. A realistic specimen of

a composite superconductor, however, would be very difficult and expensive

to fabricate. An alternate method to evaluate G experimentally is

outlined below.

From the two-dimensional anisotropic stress-strain relation, we

have

Sll S12 S16 '0

<e0+Tr/2 ) S12 S22 S26 JQ(2)

'e sLS16 S26 S66J •n>,

where E^ and en , /.-, are the axial strains of x' and z' axes. These lie0 0 + TT/2

in the xz plane and make angles 6 and 9 + tt/2 with the x-axis respec

tively; Yfl is the shear strain of x' and z' axes, and Cq, Oq + <2 anc*TQ are the corresponding normal and shear stresses respectively (see

Figure 11). The matrix [S] in Eq. (2) represents the compliance matrix

of the composite material in x'z' directions. In a simple tension test

with the applied load and the fibers oriented along the x direction, Eq

and e„ , ,o can be measured directly and YQ can be computed, using the6 + TT/2 t)

data obtained from strain gage rosette readings. For axial tension,

0-q, ae + 7T/2 and T0 are given by

Or, = O COS Q0 X

70-HT/2 a sin 0

T = -a sinQcos©0 x

(3)

where a is equal to the applied force divided by the cross sectional

area of the specimen. The elastic compliances S--, S10, S„„, S..,, S„.,11 ±2. 22 lo 2b _

and S are related to E , E22, v,2, and G by the following relations.

- _cos40 / 1 V12 \ .2„ 2„ ,sin4©SU " ~E +2 G ~ sinGcos0+- (4)

^11 \ 12 bll / E22

S"1? =(lT- +IT- ~7r~) sin20cos20 --i2- (sin40 +cos40) (5)V^ll 22 °12/ Ell

?«= (t+ft -4)sinecos3e +(t -ft -4)sin3ecose <7)

?26" fe+ft •4)sin3ecose +(4" ft •t)sinecos3e <8)

$66 "(4+4 +"ft "4) sl"2ecos2e +25- <sln4e +«-4e> WSince E^, E22> and v^ are determined from tests (2.1) and (2.2), and

e9> e0 + tt/2' ^0» °Q> °Q + tt/2' anci T8 are °^tained either from directmeasurement by strain gages or from Eq. (3), the only unknown in Eq. (2)

is G,2. Thus G „ can be computed from any one of the three equations in

Eq. (2). Its value is found to be 44.8 GPa which is within 5% of the

theoretical predictions (see Fig. 12).

CONCLUSIONS

The goal of this experiment was to determine which theory of com

posites best predicted the elastic mechanical behavior of a supercon

ducting (NbTi/Cu) composite wire. Examination of each elastic mechanical

property reveals that all theories examined are capable of predicting

experimental data to within 5%.

Since the Poisson's ratios for both the fiber and the matrix material

were very close, there was essentially no (less than 1%) difference among

all the theoretical predictions for any individual mechanical constant.

Because of the expense and difficulty of producing elastic constant data

within 0.1% accuracy, and therefore, conclusively determining which

theory is best, no further experiments were performed.

In conclusion, for a superconducting composite wire, NbTi/Cu, a

simple, fast, and reliable engineering estimate of its elastic mechanical

behavior can be made by using the "rule of mixtures." It is unnecessary

to use one of the more rigorous theories.

REFERENCES

1. C. W. Fowlkes et al., "Characterization of a Superconducting CoilComposite," NBSIR-73-349, Boulder, Colorado (1973).

2. J. E. Ashton, J. C. Halpin, and P. H. Petit, Primer on CompositeMaterials: Analysis, Technomic Publishing Co., Stamford,Connecticut (1969).

3. A. V. Hershey, "The Elasticity of an Isotropic Aggregate of Anisotropic Cubic Crystals," J. of Appl. Mech. 21, 235 (1954).

4. E. Kroner, "Berechnung der Elastischen Konstanten des Vielkristallsaus den Konstanten des Einkristalls," Z. Phys. 151, 504 (1958).

5. R. Hill, "Theory of Mechanical Properties of Fiber-strengthenedMaterials Self-consistent Model," Journal of Mechanics and Physicsof Solids 13, 189 (1965).

6. H. Frohlich and B. Sack, "Theory of the Rheological Properties ofDispersion," Proc. R. Soc. Lond. A185, 415 (1946).

7. J. J. Hermann, "The Elastic Properties of Fiber Reinforced MaterialsWhen the Fibers are Aligned," Proceedings Koninklijke NederlandsAkademic van Westenschappen, Amsterdam Series B. _7_P_(1), 1 (1967).

8. Z. Hashin, "Assessment of the Self-consistent Scheme Approximation -Conductivity of Particulate Composites," Journal of CompositeMaterials 2, 284 (1968).

9. Z. Hashin and W. Rosen, "The Elastic Moduli of Fiber-ReinforcedMaterials," J. of Appl. Mech. 31, 223 (1964).

10. Z. Hashin, "On Elastic Behavior of Fiber-Reinforced Materials ofArbitrary Transverse Phase Geometry," Journal of Mechanics andPhysics of Solids 13, 119 (1965).

11. R. A. Schapery, "Thermal Expansion Coefficients of CompositeMaterials Based on Energy Principle," Journal of CompositeMaterials 2/3), 380 (1968).

12. R. Hill, "Theory of Mechanical Properties of Fiber-StrengthenedMaterials: Elastic Behavior," Journal of Mechanical Physics ofSolids 2, 199 (1964).

13. D. F. Adams and D. R. Doner, "Longitudinal Shear Loading of aUnidirectional Composite," Journal of Composite Materials 1, 152(1967).

10

11

14. D. F. Adams and D. R. Doner, "Transverse Normal Loading of aUnidirectional Composite," Journal of Composite Materials 1, 152(1967).

15. C. H. Chen and S. Cheng, "Mechanical Properties of Fiber-Reinforced Composites," Journal of Composite Materials _1,30 (1967).

16. E. Behrens, "Thermal Conductivity of Composite Materials,"Journal of Composite Materials 2/1), 2 (1968).

17. S. I. Kang and G. M. Rentzepis, "On the Determination ofPhysical Properties of Composite Materials by a Three-Dimensional Finite Element Procedure," Composite Materials:Testing and Design, American Society for Testing and MaterialsSpecial Technical Publication S46, Philadelphia, Pennsylvania(1973).

18. R. M. Barker, F. T. Lin and J. R. Dana, "Three-Dimensional Finite-Element Analysis of Laminated Composites," Computers andStructures 2, 1013-1029 (1972).

19. R. M. Jones, Mechanics of Composite Materials, McGraw-Hill,New York (1974).

20. J. C. Ekvall, "Elastic Properties of Orthotropic MonofilamentLaminates," paper 61-AV56 presented at American Society ofMechanical Engineers Aviation Conference, Los Angeles, California(March 1961).

21. G. S. Springer and S. W. Tsai, "Thermal Conductivity of Unidirectional Materials," Journal of Composite Materials _1, 166(1967).

22. J. C. Halpin and S. W. Tsai, "Environmental Factors in CompositeMaterials Design," AFML-TR-67-423, Air Force Materials Laboratory,(1967).

23. G. A. Miranda and J. D. Rogers, "Adjacent Conductor Field Corrections to High Critical Current Short Sample Measurements,"LA-UR-74-1413, Los Alamos, New Mexico (1974).

24. B. W. Rosen, "A Simple Procedure for Experimental Determinationof the Longitudinal Shear Modulus of Unidirectional Composites,"Journal of Composite Materials 6^, 552 (October 1972).

25. J. M. Whitney, "Geometrical Effects of Filament Twist on theModulus and Strength of Graphite Fiber-Reinforced Composites,"Textile Research Journal 36(9), 765 (September 1966).

NOMENCLATURE

Symbol Definition

E - Young's modulus

G - Shear modulus

V - Poisson's ratio

K - Bulk modulus in plane strain

V - Volume fraction

f - Fiber material (NbTi)

m - Matrix material (Cu)

L - Composite material constants along the fiber direction

T - Composite material constants along a transverse direction

12

APPENDIX I

This appendix presents the various theoretical equations which were

used to predict the elastic mechanical properties for the superconducting

composite wire studied in this report. Their alphabetic index refers to

the legend on the particular graph where a comparison with experimental

data was performed. Also included in this appendix is the reference

in which these equations may be found.

A program which numerically tabulates all of these equations is

listed in Appendix III.

13

14

1. Longitudinal Young's modulus E.

(a) ET = E V + ECVC (10)L mm f f

Source: Reference 2

(b) EL - (Vm +VfJ * (IDEm (Dl " °3Fl) + Ef(°2 " °4F2)

(f, =E (D, - D.) + E,(D, - D.)mv 1 3' fv 2 4'

1 + V,

D = 1 - V, D = —-—- + V1 f 2 V m

m

V2 2 f

D, = 2v, D. = 2V V3 f 4 m V

m

V VÊ£ + V^V E_ m f f f m m

1 V.E.VC + V E V.f f f m m f

VfF = —- F2 V 1

m

Source: Reference 10

15

2. Major Poisson's ratio VL

V-E.L.. + V E V L_f f 1 m m m 2 /10.

(a) VL "VfEfL, HEL, (12)f f 3 m m 2

Ll " 2VfI1 "Vm2)Vf +VmlX +VmjVmL2 =I1- vf - 2vf2)vf

(1 +VV

L3 =2l1_V-2)Vf +(1 +Vtn)VmSource: Reference 10

(b) VT = V V + v,V, (13)L mm if

Source: Reference 2

16

3. Transverse Young's modulus E

E,Ef m

(a) ET =T E£V + E V,f m m f


i + 4>nv.(b) E,, =T 1 - r|Vf m

m

Ef

m

n =

<j) = 2 for circular fiber

<)) = 2It-) for rectangular fiber

bC I 1 E„, = ET x

Source: Reference 2

(c) ET =Mf(2Mm + Gj - Gm(Mf - Mjvfm m ml mm

(. ri _ V ^.n t u j — u Vile J.1 V\ i fv m my mv f m'

Vf " Vm m 2M + G + 2(M, - M)V/ J m m f mm

mM„ =f 2(1 - Vj

M =m 2(1 - V )


(14)

(15)

(16)

(d) ^ =m

-2l^) +if tt / a

a = 21 -S-l

17

h-



V V, V, \ E Vm " Vf

^ ET_Em Ef"Ef VfV E

m m

+ 1

tan"

a-a2v, 1+|/*Vrr> (17)

•

(18)

18

4. Transverse shear modulus Gn

(a) Gm (lower bound) = GT m

2<! " VJ1 1 - 2 V f 4

m

where A. is obtained by solving the following4

equations

Ai i

[p] < >=<

0

0

0

0

0

v. -/

where

[P] =

^2,

,-1

0

.. 4V -31 m

V£ 3 - 2Vf m

1 1

0

4v -3m

3 - 2Vm

13

3 - 2vm

n1

3 - 2vm


V'

-2V,

-2

-3

V,

1 - 2Vm

1 - 2vm

1 - 2vm

-1

1 - 2vm

-1

0

ffG

0

-1

3 - 4v,

3 - 2vf

3 Gf/Gmr m

2vf - 3

G./Gf m

3 - 2vf J

(19)

19

(b) G (upper bound) = G2(1 " VJ a

1 + -. T-2 V,Aa

and A, is obtained by solving

[R],

where

[R] =

>

r *sr *\

A, 11

A2 0

A3 0

A? \-\ 0 14

R, 01

B2 0^ J

3 1

3 - 2v V,m f

-1 1

TT- -3V

3 - 2v V,m f

4V -3m

3 - 2vm

3

3 - 2Vm

1

3 - 2vm

2V

-2

-3


1 - 2v f 4m

1 - 2vm

-V,

1 - 2vm

1

1

1 - 2Vm

1

1 - 2vm

-1

1 - 2Vm

-1

0

ffGm

-1

3 - 4vf

3 - 2vf

3 Gf/Gm1 m

2vf - 3

Gf/G™i m

3 - 2v^

(20)

20

2GJK + G V£ + 2G G„V + K V G + G,f \ m m/ f m f m m mVm zj(C) GT = Gm 2G Ac +G V, + 2GGJ + KV A3 + G,\ (21)

™lm mf mfm mmm f' v/

Source: Reference 7

Vf(d) G (lower bound) = G + (K + 2G V ) (22)

1 m m mG, - G + 2G (K + G )

f m mmm


V

(e) GT (upper bound) = Gf + ——rj-r^—rr- (23)

G7^+2Gf(Kf + Gf)


21

5. Longitudinal shear modulus G

G, (1 + Vf) + Vt r m

(a) GT =L G,

-i- V + 1 + VG m 1m


G,Gf m

(b) GT =L V G. + V,G

m f f m


(c) GT =i + (|>nvfl - nvf m

m 7 x m

+ - t;ft

for rectangular fiber

I—I

I 1

4 y

(j) = 1 for circular fiber

Source: Reference 2

(24)

(25)

(26)

GL " GXZ

(d) G, =2G, - (G - G ) V

1 1mm

L 2G + (Gr - G ) V mm f m m


(e) GT (lower bound) = G +L m

22

1 + mG, - G 2Gf m m


(f) G (upper bound) = Gf +m

1 + fG - G. 2G„m f 1


(g) GL= 2m 4 - TT + TT(j)+ ±

(J) (4 - TT) + TT

* =

Gc (tt + 4V-) + (G tt - 4V.)f 1 m r__

G. (tt - 4V.) + (G tt + 4V.)f f m f


(27)

(28)

(29)

(30)

APPENDIX II

Whitney has investigated the influence of twist on graphite fibers

in an epoxy matrix. He derived an equation for the reduction of the

longitudinal elastic Young's modulus for graphite as a function of the

geometry of the fibers. This equation is directly applicable in esti

mating how twisting affects a superconducting wire.

If the initial and reduced moduli are E and ER, respectively, then25

their ratio can be expressed as :

ER

EI 1 + 4tt2N2R2 ,o '

where N is the number of twists per centimeter and R is the radius ofo

the fiber in centimeters.

For the superconducting wire analyzed in this report,

N = .132 cm"1o

R = 3.17 x 10~3cm,

which yields

1 + 4tt2N2R2 1 + 6.87 x 10~6o

1.

Clearly, the effect of twist for this superconducting wire can be

neglected.

23

APPENDIX III

25

FFFFFFFFFFFFFFF IIII IIII BBBBBBBBBBBE EEEEEEEEEEEEEEE RRRRRRRRRRRR CCCCCCCCCCCC



FFF BBB BBB EEE RRR RRR CCC

FFF BBS BBB EEE RRR RRR CCC





FFFFFFFFFFFF BBBBBBBBBBBE EEEEEEEEEEEE RRRRRRRRRRRR CCC









FFF IIII IIII BBBBBBBBBBBB EEEEEEEEEEEEEEE RRR RRR CCCCCCCCCCCC

FFF IIII IIII BBBBBBBBBBBE EEEEEEEEEEEEEEE RRR RRR CCCCCCCCCCCC

FFF IIII IIII BBBBBBBBBBBB EEEEEEEEEEEEEEE RRR RRR CCCCCCCCCCCC

vw wv AAAAAAAAA

wv vw AAAAAAAAA

vw vw AAAAAAAAA

vw vw 1111 AAA AAAwv vw 1111 AAA AAAvw vw mi AAA AAAvw vw AAA AAAvw wv AAA AAA

wv vw AAA AAA

vw wv AAA AAA

wv vw AAA AAA

vw vw AAA AAAwv wv AAAAAAAAAAAAAAAwv wv AAAAAAAAAAAAAAAvw wv AAAAAAAAAAAAAAA

wv wv AAA AAA

wv wv AAA AAAwv vw AAA AAA

wv 1111 mi AAA AAA

wv 1111 mi AAA AAA

wv mi 1111 AAA AAA

LPTSPL Version 6(34404) Running on EPL000*START* User GRAY C200.2304] Job FIBERC Seq. 6417 Date 17-May-76 16:03:00 Monitor ORNL-USC KI-10 NET 602 *START*Request croated: 17-May-76 16:04:07File: DSKB0:FIBERC.V1AC200.23043 Created: 02-Apr-76 10:40:00 <157> Printed: 17-May-76 16:03:17QUEUE Switches: /PRINT:ARROU /"FILE:ASCII /COPIES: 1 /SPACING: 1 /LIMIT:34 /FORMS:NORMALFile will be RENAMEd to <057> protection

S3

CJ\

27

cC PROGRAM FIBERC.VIACC LAST UPDATED: 1APR76CC AUTHOR: U. H. GRAYC P. 0. BOX Y 9204-1C OAK RIDGE NATIONAL LABORATORYC OAK RIDGE, TN. 37B30

CC LANGUAGE: DECSYSTEM-10. FORTRAN-10CC SUBROUTINES REQUIRED:CC MINV - STANDARD IBM SCIENTIFIC SUBROUTINE PACKAGE.C

IMPLICIT REAL <K)REAL NUF-NUMDIMENSION U(6.6).RES(6).L(6).M(6).A(6)DIMENSION UINVC6.6)

• DIMENSION ELEM(2),XNULT(2).GLGM(7).GTGM(5),1 KTKM(4).ETEM(5)

DATA PI/3.1415926/DATA IORED.IOURT/5.5/

CC BEGIN PROGRAM FIBERC.VIACC QUERY THE USER FOR THE MATERIAL PROPERTIES

URITEUOURT. 11)11 FORMATC//' FIBERC.VIA DOCUMENTED IN 0RNL/TM-533r/

1 ' INPUT IN THE FOLLOWING ORDER:'/2 ' EF(FIBER YOUNGS MODULUS)'/3 * EM(MATRIX YOUNGS MODULUS)'/4 ' NUF(FIBER POISSONS RATIO)'/5 ' NUM(MATRIX POISSON RATIO)'/6 ' VF (FIBER VOLUME FRACTION)'//7 ' ORDER IS EF.EM,NUF,NUM,VF F0RMAT(5F)*/B ' REMEMBER A SPACE DELIMITS THE INPUT VARIABLES'/)

READUORED. 10)EF.EM,NUF.NUM, VF10 FORMAT(SF)989 CONTINUECC CALCULATE THE SHEAR MODULI I

GF-EF/(2.*(t.+NUF))GM-EM/(2.*(1.-HHUM))

C CALCULATE THE MATRIX VOLUME FRACTIONVM-l.-VF

CC CALCULATE THE BULK MODULI I

KF-EF/(3.*(1.-2.*NUF)>KM-EM/(3.*(1.-2.*NUM))

CC CALCULATE THE FIBER-MATRIX RATIOS.

GMGF-GM/GFGFGM-GF/GMEFEM-EF/EMEMEF-EM/EFGMEM-GM/EMKMGM-KM/GMKFGF-KF/GF

28

KFGM-KF/GMC ECHO INPUT DATA

URITE(I0URT.29) EF.EM.NUF.NUM.VF.GF.GM.EFEM.KF.KM.GFGM29 FORMAT(//' INPUT DATA'//.

1* EF - '.T10.1PE11.4,/.2' EM - '.T10.1PE11.4./.3' NUF - '.TIB. 1PE11.4./.4' NUM - '.TIB. 1PE11.4./,5' VF - '.T10.1PE11.4./.6' GF - '.T10.1PE11.4./.7' GM - '.T10.1PE11.4./.8' EF/EM - '.T10.1PE11.4./.9' KF - '.T10.1PE11.4./..' KM - '.T10.1PE11.4./.' GF/GM - ' .T10.1PE11.4)

CC THE EQUATION NUMBERS GIVEN IN THE REST OF THIS CODE REFERC TO ORNL/TM-5331

CC CALCULATE THE LONGITUDINAL YOUNG'S MODULUSCC EQUATION 10

ELEM(1)-VF*EFEM+VMC EQUATION 11

IF(VM.NE.0.0) GO TO 21ALPHA-1.GO TO 20

21 CONTINUEF1-(NUM*VF*EFEM+NUF*VM)/(NUF*VF*EFEM+NUF*VM)F2-NUF*F1/NUMDl-l.-NUFD2-(1.+VF)/VM+NUMD3-2.*NUF*NUFD4-2.*NUM*NUM*VF/VMALPHA-D1-D3*F1+EFEM*(D2-D4*F2)ALPHA-ALPHA/(D1-D3+EFEM*(D2-D4))

28 ELEM(2)-ELEM(1)*ALPHAURITEU0URT.25)

25 FORMAT(//.' LONGITUDINAL MODULUS'/)C OUTPUT DATA ON DATA FILE

DO 801 J-1.2801 ELEM(J)-ELEM(J)*EM

DO 30 J-1.2JJ-J+9URITE(IOURT.24) JJ.ELEM(J)

30 CONTINUE24 FORMATdX.'EQU. '. I2.4X. 1PE11.4)CC CALCULATE THE MAJOR POISSON RATIOCC EQUATION 12

XL 1 -2. *NUF* (1. -NUM*NUM) *VF+NUM* (1. -HftJM) *VMXL2-VF*(1.-NUF-2.*NUF*NUF)XL3-2.*C1.-NUM*NUM)*VF+(1.+NUM)*VMXNULTA-VF*EFEM*XL1+VM*NUM*XL2XNULT(l)-XNULTA/(VF*EFEM*XL3+VM*XL2)

CC EQUATION 13

XNULTC2)-VF*NUF+VM*NUMURITECIOURT, 225)

225 FORMAT(//,' MAJOR POISSONS RATIO'/)

29

C OUTPUT DATA ON DATA FILEDO 31 J-1.2JJ-J+11URITEdOURT. 24) JJ.XNULT(J)

31 CONTINUECC CALCULATE THE TRANSVERSE YOUNG'S MODULUSCC EQUATION 14

ETEM(1)-1./(VM+EMEF*VF)

CC EQUATION 15

ETA-(EFEM-l.)/(EFEM+2.)ETEM(2)-(1.+2.*ETA*VF)/(1.-ETA*VF)

CC EQUATION 16

XMF-EFEM/(2.*(1.-NUF))XMM-1./(2.*(1.-NUM))PART-2.*(1.-NUF+(NUF-NUM)*VM)PART-PART*(XMF*(2. *XMM+GMEM) -GMEM*(XMF-XM1)*VM)ETEM(3)-PART/(2.*XMM+GMEM+2.*(XMF-XMM)*VM)

CC EQUATION 17

ALPHA-2.*(EMEF-1.)CC CHECK FOR A POSITIVE RADICAL FOR SORT.

IF(1.-ALPHA*ALPHA*VF/PI.LE.0.) GO TO 160XNMR-SQRT(1.-ALPHA*ALPHA*VF/PI)TANSTF-ATAN2(XNMR.(1.+SQRT(ALPHA*ALPHA*VF/PI)))SQ-4./SQRT(1.-ALPHA*ALPHA*VF/PI)ETEM(4)-1.-2.*SQRT(VF/PI)+(PI-SQ*TANSTF)/ALPHAGO TO 161

160 CONTINUEETEMU) — 1.

161 CONTINUECC EQUATION 18

SQ-(EFEM*NUM-NUF)**2*VF/EFETEM(5)-VM/EM+VF/EF-SQ/(VF*£F/VM/EM+1.)ETEM(5) -1. /(ETEM(5) *EM)URITEdOURT. 625)

625 FORMAT(//,' TRANSVERSE YOUNGS MODULUS*/)C OUTPUT DATA ON DATA FILE

EHG-AMAXKEF.EM)ELU-AMINKEF.EM)DO 8B5 J-1.5

805 ETEM(J)-ETEM(J)«EMDO 32 J-1.5JJ-J+13IF(ETEM(J).GT.EHG.OR.ETEMU).LT.ELU) GO TO 82URITEdOURT.24) JJ.ETEMU)GO TO 32

82 URITEdOURT.83) JJ83 FORMATC EQU. '.12.' NOT APPLICABLE')32 CONTINUECC CALCULATE THE TRANSVERSE SHEAR MODULUSCC EQUATION 19C

30

C BEGIN TO SET UP THE COEFFICIENT MATRIXU(l.l)-1.U(2.1)-0.U(3.1) = l.U(4.1)=8.U(5.1)-l.U(6. l)-0.

U(3.2)-l.U(3.3)-l.U(4,3) — 2.

U(5.3)—3.U(6.3)-2.U(3.4)-l.U(1.5)-0.U(2.5)-8.U(3,5) — 1.U(4.5)-0.U(6.5)-0.U(1.6)-0.U(2.6)-0.U(3.6) —1.

U(1.2)-1./VFU(4.2)—(3.-4.«NUM)/(3.-2.*NUM)U(2.2)-U(4,2)/VFU(6.2) — l./(3.-2.*NUM)U(5.2)—3.*U(6.2)U<1.3)-VF*VFU(2.3)—2.*U(1.3)U(1.4)-VF

U(4.4)-1./(1.-2.*NUM)U(2.4)-VF*U(4.4)U(5.4)-U(4.4)U(6.4)—U(4.4)U(5.5)—GFGM

U(4,6)-(3.-4.*NUF)/(3.-2.*NUF)U(6,6)-GFGM/(3.-2.*NUF)U(5.6)—3.*U(6.g)

C

C STORE THE COEFFICIENT MATRIX INTO UINVDO 71 IH-1.6DO 71 JH-1.6

?l UINV(IH.JH)-UdH.JH)C

C FORMULATE RIGHT HAND SIDEAd)-1.A(2)-0.A(3)-0.A(4)-0.A(5)-0.A(6)-0.

C

C INVERT THE COEFFICIENT MATRIX. SUBROUTINE MINV IS THE STANDARDC IBM SCIENTIFIC SUBROUTINE MATRIX INVERTER. STANDARD

CALL MINV(UINV.6.D.L.M)C

C OBTAIN THE SOLUTION VECTORDO 69 MK-1.6RES(MK)-0.DO 69 ML-1.6

69 RES<MK)-A(ML)*UINV(MK.ML)+RES(MK)

31

GTGM(1)-1.-2.*(1.-NUM)*VF*RES(4)/(1.-2.*NUM)Cccc

EQUATION 20

BEGIN TO SET UP THE COEFFICIENT MATRIXU(l.l)-1.U(2.1)-0.

U(3.1)-l.U(4. l)-0.U(5.1)-l.U(6,l)-0.U(1.2)-l.Ud.3)-1.U(2.3)—2.U(3.3)—3.U(4.3)-2.U(1.4)-l.U(1.5) —1.U(2.5)-0.U(4.5)-0.U(5.5)-0.

U(6.5)-0.U(1.6) —1.U(5.6)-0.U(6.6)-0.U(4.2) —l./(3.-2.*NUM)U<3.2)—3.*U<4.2)U(2.2)-(3.-4.*NUM)*U(4.2)U(5.2)-U(3.2)/VFU(6.2)-U(4.2)/VFU(5.3)—3.*VF*VFU(6.3)-2.*vF*VFU(2.4)-1./(1.-2.*NUM)U(3.4)-U(2.4)U(4.4)—U(2.4)U(5.4)-VF*U(2.4)U(6,4)—U(5.4)U(3.5)—GFGMU(2.6)-(3.-4.*NUF)/(3.-2.*NUF)U(4.6)-GFGM/(3.-2.*NUF)

cc

U(3.6)—3.*U(4.6)

STORE THE COEFFICIENT MATRIX INTO UINVDO 81 IH-1.6DO 81 JH-1.6

81 UINVdH.JH)-U(IH.JH)cC FORMULATE THE RIGHT HAND SIDE.

Ad)-0.A(2)-0.A(3)-0.A(4)-0.

A(5)-l.A(6)-0.

Cc INVERT THE MATRIX.

cc

CALL MINV(UINV.6.D.L.M)

OBTAIN THE SOLUTION VECTOR.DO 89 MK-1.6

32

RES(MK)-0.DO 89 ML-1.6

89 RES(MK)-A(ML)*UINV(MK.ML)+RES(MK)GTGM(2)-1./(1.+2.*(1.-NUM)/(1.-2.*NUM)*VF*RES(4))

CC EQUATION 21

KMGM1-(KMGM+1.)*VF*2.GFGM1-(GFGM+1.)*VM*KMGMGTGMC-GFGM*KMGM1+2.*GFGM*VM+GFGM1GTGM(3)-GTGMC/(KMGM1+2.*GFGM*VM+GFGM1)

CC EQUATION 22

GTGM(4)-1.+VF/(1./(GFGM-1.)+(KMGM+2.)*VM/(2.*(KMGM+1.)))CC EQUATION 23

FRAC-((KFGF+2.)*VF/2./(KFGM+GFGM)+1./(1.-GFGM))108 GTGM(5)-GFGM+VM/FRAC

URITEdOURT. 425)425 FORMAT(//,' TRANSVERSE SHEAR MODULUS'/)C OUTPUT DATA ON DATA FILE

DO 803 J-1.5803 GTGMU)-GTGM(J)*GM

DO 33 J-1,5JJ-J+18URITEdOURT.24) JJ.GTGM(J)

33 CONTINUECC CALCULATE THE MINOR POISSON RATIO USING THE RULE OF MIXTURESC TRANSVERSE YOUNG'S MODULUS AND EACH OF THE PREVIOUSLY CALCULATEDC TRANSVERSE SHEAR MODULI I.C

E22-ETEM(1)URITEdOURT, 79)DO 75 J-1.5JJ-J+1B

CC EQUATION 1

P0I23-(E22/(2.*GTGM(J)))-1.IF(PDI23.LT.0.0.OR.POI23.GT.0.5) GO TO 76URITEdOURT.77) JJ.P0I23GO TO 75

76 URITEdOURT.78) JJ75 CONTINUE7? FORMATC EQU. 1 USING EQU. '. I2.4X. 1PE11.4)

78 FORMATC EQU. 1 USING EQU. '.12.' NOT APPLICABLE')79 FORMAT(//.' MINOR POISSONS RATIO'/)CC CALCULATE THE LONGITUDINAL SHEAR MODULUSCC EQUATION 24

GLGMd) -(GFGM*(1. +VF) +VM)/(GFGM*VM+1 .+VF)

CC EQUATION 25

GLGM(2)-1./(VM+GMGF*VF)CC EQUATION 26

ETA-VF*(GFGM-1.)/(GFGM+1.)GLGM(3)-d.+ETA)/d.-ETA)

CC EQUATION 27

33

GLGMD-2.*GFGM-(GFGM-1.)*VMGLGM(4)-GLGMD/(2.+(GFGM-l.)*VM)

CC EQUATION 28

FRAC-1./(GFGM-1.)GLGM(5)-1.+VF/(FRAC+VM/2.)

C

C EQUATION 29FRAC-1./(1.-GFGM)+VF/(2.*GFGM)GLGM(6)-GFGM+VM/FRAC

C

C EQUATION 30

ALPHA-GFGM*(4.*VF+PD+PI-4.*VFALPHA-ALPHA/(GFGM*(PI-4.*VF)+PI+4.*VF)

UR?TE?lOl!RTa325PIÎ^LPHft)/4'+4',WLPHfl''(PLPHft*C4'"PI)'H3I))325 FORMAT(//.''LONGITUDINAL SHEAR MODULUS'/)C OUTPUT DATA ON DATA FILE

GHG-AMAxl(GF.GM)GLU-AMINKGF.GM)DO 802 J-1.7

802 GLGMU)-GLGMU)*GMDO 34 J-1.7JJ-J+23IF(GLGM(J).GT.GHG.OR.GLGM(J).LT.GLU) GO TO 85URITEdOURT.24) JJ.GLGM(J)GO TO 34

85 URITEdOURT.83) JJ34 CONTINUECC MORE DATA

URITEdOURT. 990)990 FORMATC//' MORE DATA?'/

1 ' INPUT EF.EM.NUF.NUM.VF F0RMATC5F)'/2 ' NEGATIVE EF STOPS THE PROGRAM')

READ(IORED.10)EF.EM.NUF. NUM.VFIF(EF.GT.0.) GO TO 989STOPEND

ACKNOWLEDGMENTS

The authors wish to thank. S. E. Bolt for his assistance with the

experimental aspect of this paper and C. J. Long and W.C.T. Stoddart

for their useful comments during the preparation of this manuscript.

We also wish to thank K. E. Rothe for her assistance with the computer

program listed in Appendix III.

35

FIGURE CAPTIONS

Fig. 1. Direction of loading for the determination of the longitudinalYoung's modulus and the major Poisson's ratio.

Fig. 2. a vs e diagram for Kryo-210 superconductor.

Fig. 3. Normalized plot of the longitudinal Young's Modulus versusvolume fraction of the fiber (NbTi): In this graph, aswell as the other normalized comparison graphs, the legendrefers to the theoretical equations presented in Appendix I.

Fig. 4. e2 vs e diagram for Kryo-210 superconductor.

Fig. 5. Normalized plot of the major Poisson's ratio vs volumefraction of the fiber (NbTi).

Fig. 6. Direction of loading for the determination of the transverseYoung's Modulus and the minor Poisson's ratio.

Fig. 7. a2 vs e„ diagram for Kryo-210 superconductor.

Fig. 8. Normalized plot of the transverse Young's Modulus versusvolume fraction of the fiber (NbTi).

Fig. 9. e^ vs e2 diagram for Kryo-210 superconductor.

Fig. 10. Normalized plot of G vs the volume fraction of thefiber (NbTi). ZJ

Fig. 11. Coordinate system for the determination of the longitudinalshear modulus.

Fig. 12. Normalized plot of the longitudinal shear modulus vs volumefraction of the fiber (NbTi).

36

37

SUPERCONDUCTOR (NbTi) FILIMENT

//

Figure 1

Q.

Ib~

38

i i

50 -

40

30

20SLOPE = E =119 GPa

10

.î i i i i i i 1 1 1 ».

100

e(~ ^l m/m

Figure 2

200 300

39

ORNL DWG. 75-16576

1.00

Figure 3

40

20 40 60 20O 300

Figure 4

41

ORNL DWG. 75-16577

1.00

Figure 5

42

cr2 SUPERCONDUCTING (NbTi) FILIMENT

i2

/

Figure 6

43

€2~ f± m/m

Figure 7

OCD

^.0- KRYO-210

0.25

44

0.50

Vf

Figure 8

ORNL DWG. 75-16575

0.75 1.00

E

E

ro

60

40

20

0

100

45

23"2(l* I/23) = 43.1 GPa

J L

€2~ jJL m/m

Figure 9

200

OCO

46

- KRYO-210

0.50

VfFigure 10

ORNL DWG. 75-16573

47

N

A

V

Figure 11

O

^).0'- KRYO-210

0.25

48

0.50

VfFigure 12

ORNL DWG. 75-16574

0.75 1.00

INTERNAL DISTRIBUTIONITION

0RNL/TM-5331

33. J. R. Miller

34. 0. B. Morgan35. H. Pih

36. H. Pos tma

37. M. Roberts

38. M. W. Rosenthal

39. R. E. Schwall

40. T. E. Shannon

41. S. S. Shen

42. J. E. Simpkins

43. D. Steiner

44. w. C. Stoddart

45. P. B. Thompson46. P. L. Walstrom

47. H. T. Yeh

48. A. Zucker

49-50. C. R. Library (2)51. CTR Reports Office

52. Document Ref. Section

53-54. Lab Records Dept. (2)55. Lab Records, ORNL, R.C

56. Patent Office

57-58. Thermonuclear Division

Library (2)

1. W. C. Anderson

2. J. K. Ballou

3. R. L. Brown

4. E. H. Bryant

5. P. B. Burn

6. w. D. Cain

7. D. D. Cannon

8. J. F. Clarke

9. F. L. Culler

10. L. Dresner

11. J. F. Ellis

12. W. A. Fietz

13. A. P. Fraas

14. K. J. Froelich

15-22. W. H. Gray

23. P. N. Haubenreich

24. G. G. Kelley

25. C. G. Lawson

26. C. J. Long

27. H. M. Long

28. J. K. Lovin

29. M. s. Lubell

30. J. W. Lue

31. J. N. Luton

32. H. C. McCurdy

EXTERNAL DISTRIBUTION

59. E. Adam, Airco, 100 Mountain Avenue, Murray Hill, NJ 07974.60. J. W. Beal, DCTR, ERDA, Mail Stop G-234, Washington, DC 20545.61. R. W. Boom, 513 Engineering Research Building, University of

Wisconsin, Madison, WI 53706.62. D. L. Coffey, American Magnetics Inc., P.O. Box R, Oak Ridge, TN 3783063. D. N. Cornish, Lawrence Livermore Laboratory, P.O. Box 808,

Livermore, CA 94550.64. M. D'Agostino, Grumman Aerospace Corp., Bethpage, NY 11714.65. J. E. Darby, Jr., CTR Program, D208, Argonne National Laboratory,

9700 South Cass Avenue, Argonne, IL 60439.66. R. W. Derby, 247 Third Street, Cambridge, MA 02141.67. Director, Research & Technical Services Division, ERDA, Oak Ridge

Operations, Oak Ridge, TN 37830.68. R. W. Fast, Manager, Experimental Facilities, National Accelerator

Laboratory, P.O. Box 500, Batavia, IL 60510.69. J. J. Ferrante, General Electric Co., Schenectady, NY 12345.70. J. File, Princeton University, Plasma Physics Laboratory,

Princeton, NJ 08540.71. W. S. Gilbert, Lawrence Berkeley Laboratory, University of California,

Berkeley, CA 94120.

49

50

72. R. W. Gould, Dept. of Applied Physics, California Institute ofTechnology, Pasadena, CA 91109.

73. H. Grad, Courant Institute, New York University, 251 Mercer Street,New York, NY 10012.

74. E. Gregory, Airco, 100 Mountain Avenue, Murray Hill, NJ 07974.75. W. V. Hassenzahl, Los Alamos Scientific Laboratory, P.O. Box 1663

Los Alamos, NM 87544.76. C. D. Henning, DCTR, ERDA, Mail Stop G-234, Washington, DC 20545.77. G. K. Hess, DCTR, ERDA, Mail Stop G-234, Washington, DC 20545.78. C. K. Jones, Manager, Cryogenic Research Laboratory, Westinghouse

Electric Corporation, R&D Center, Pittsburgh, PA 15235.79. E. E. Kintner, DCTR, ERDA, Mail Stop G-234, Washington, DC 20545.80. H. L. Laquer, Los Alamos Scientific Laboratory, P.O. Box 1663,

Los Alamos, NM 87544.81. C. Laverick, 543 Hampshire Lane, Boling Brook, IL 60439.82. J. A. Mayhall, Lockheed, P.O. Box 1103, Huntsville, AL 35807.83. D. B. Montgomery, MIT, National Magnet Laboratory, 170 Albany

Street, Cambridge, MA 02139.84. F. Moon, Dept. of Theoretical & Applied Mechanics, Cornell University,

Ithaca, NY 14850.85. R. W. Moses, Jr., University of Wisconsin, Madison, WI 53706.86. J. R. Powell, Brookhaven National Laboratory, Upton, NY 11973.87. J. R. Purcell, General Atomic Company, P.O. Box 81608, San Diego,

CA 92138.

88. J. D. Rogers, Q-26, Los Alamos Scientific Laboratory, P.O. Box1663, Los Alamos, NM 87544.

89. D. J. Rose, Dept. of Nuclear Engineering, Massachusetts Instituteof Technology, Cambridge, MA 02139.

90. C. H. Rosner, Intermagnetics General Corp., Charles IndustrialPark, New Karner Road, Guilderland, NY 12084.

91. W. B. Sampson, Brookhaven National Laboratory, Upton, Long Island,

NY 11973.

92. Z.J.J. Stekly, Magnetic Corporation of America, 179 Bear HillRoad, Waltham, MA 02154.

93. B. P. Strauss, Fermi National Accelerator Laboratory, P.O. Box

500, Batavia, IL 60510.94-120. TIC, Oak Ridge, TN 37830. (27)

121. C. E. Taylor, Lawrence Livermore Laboratory, P.O. Box 808, L-384Livermore, CA 94551.

122. R. Thomas, General Atomic Company, P.O. Box 81608, San Diego,

CA 92138.

123. S. T. Wang, Argonne National Laboratory, 9700 South Cass Avenue,

Argonne, IL 60439.124. J. M. Williams, DCTR, ERDA, Mail Stop G-234, Washington, DC 20545.125. J. Wong, Supercon, Inc., 9 Erie Drive, Natick, MA 01760.126. H. H. Woodson, Chairman, Dept. of Electrical Engineering,

University of Texas at Austin, Austin, TX 78712.127. E. J. Ziurys, Mail Stop G-234, DCTR, ERDA, Washington, DC 20545.

Theoretical and experimental determination of mechanical ...

Documents